FCLoopFindTopologyMappings
FCLoopFindTopologyMappings[{topo1, topo2, ...}] finds
mappings between topologies (written as FCTopology objects)
topo1, topo2, .... For each source topology the function
returns a list of loop momentum shifts and a GLI
replacement rule needed to map it to the given target topology. If you
need to map everything to a particular set of target topologies, you can
specify them via the PreferredTopologies option.
The output is a list of two lists, the former containing the mappings
and the latter enumerating the final contributing topologies
To enable shifts in the external momenta you need to set the option
Momentum to All.
See also
Overview , FCTopology , GLI , FCLoopFindTopologies .
Examples
Here we have a set of 5 topologies
= { [ fctopology1, { SFAD[{{ p3, 0 }, { 0 , 1 }, 1 }], SFAD[{{ p2, 0 }, { 0 , 1 }, 1 }], [{{ p1, 0 }, { 0 , 1 }, 1 }], SFAD[{{ p2 + p3, 0 }, { 0 , 1 }, 1 }], SFAD[{{ p2 - Q , 0 }, { 0 , 1 }, 1 }], [{{ p1 - Q , 0 }, { 0 , 1 }, 1 }], SFAD[{{ p2 + p3 - Q , 0 }, { 0 , 1 }, 1 }], SFAD[{{ p1 + p3 - Q , 0 }, { 0 , 1 }, 1 }], [{{ p1 + p2 + p3 - Q , 0 }, { 0 , 1 }, 1 }]}, { p1, p2, p3}, { Q }, {}, {}], [ fctopology2, { SFAD[{{ p3, 0 }, { 0 , 1 }, 1 }], [{{ p2, 0 }, { 0 , 1 }, 1 }], SFAD[{{ p1, 0 }, { 0 , 1 }, 1 }], SFAD[{{ p2 + p3, 0 }, { 0 , 1 }, 1 }], [{{ p2 - Q , 0 }, { 0 , 1 }, 1 }], SFAD[{{ p1 - Q , 0 }, { 0 , 1 }, 1 }], [{{ p2 + p3 - Q , 0 }, { 0 , 1 }, 1 }], SFAD[{{ p1 + p2 - Q , 0 }, { 0 , 1 }, 1 }], [{{ p1 + p2 + p3 - Q , 0 }, { 0 , 1 }, 1 }]}, { p1, p2, p3}, { Q }, {}, {}], [ fctopology3, { SFAD[{{ p3, 0 }, { 0 , 1 }, 1 }], [{{ p2, 0 }, { 0 , 1 }, 1 }], SFAD[{{ p1, 0 }, { 0 , 1 }, 1 }], [{{ p2 + p3, 0 }, { 0 , 1 }, 1 }], SFAD[{{ p1 + p3, 0 }, { 0 , 1 }, 1 }], [{{ p2 - Q , 0 }, { 0 , 1 }, 1 }], SFAD[{{ p2 + p3 - Q , 0 }, { 0 , 1 }, 1 }], [{{ p1 + p3 - Q , 0 }, { 0 , 1 }, 1 }], SFAD[{{ p1 + p2 + p3 - Q , 0 }, { 0 , 1 }, 1 }]}, { p1, p2, p3}, { Q }, {}, {}], [ fctopology4, { SFAD[{{ p3, 0 }, { 0 , 1 }, 1 }], [{{ p2, 0 }, { 0 , 1 }, 1 }], SFAD[{{ p1, 0 }, { 0 , 1 }, 1 }], [{{ p2 + p3, 0 }, { 0 , 1 }, 1 }], SFAD[{{ p1 + p3, 0 }, { 0 , 1 }, 1 }], [{{ p2 - Q , 0 }, { 0 , 1 }, 1 }], SFAD[{{ p1 - Q , 0 }, { 0 , 1 }, 1 }], [{{ p1 + p3 - Q , 0 }, { 0 , 1 }, 1 }], SFAD[{{ p1 + p2 + p3 - Q , 0 }, { 0 , 1 }, 1 }]}, { p1, p2, p3}, { Q }, {}, {}], [ fctopology5, { SFAD[{{ p3, 0 }, { 0 , 1 }, 1 }], [{{ p2, 0 }, { 0 , 1 }, 1 }], SFAD[{{ p1, 0 }, { 0 , 1 }, 1 }], [{{ p1 + p3, 0 }, { 0 , 1 }, 1 }], SFAD[{{ p2 - Q , 0 }, { 0 , 1 }, 1 }], [{{ p1 - Q , 0 }, { 0 , 1 }, 1 }], SFAD[{{ p1 + p3 - Q , 0 }, { 0 , 1 }, 1 }], [{{ p1 + p2 - Q , 0 }, { 0 , 1 }, 1 }], SFAD[{{ p1 + p2 + p3 - Q , 0 }, { 0 , 1 }, 1 }]}, { p1, p2, p3}, { Q }, {}, {}]} ;3 of them can be mapped to the other two
= FCLoopFindTopologyMappings[ topos1] ;FCLoopFindTopologyMappings: Found 3 mapping relations \text{FCLoopFindTopologyMappings:
}\;\text{Found }3\text{ mapping relations } FCLoopFindTopologyMappings: Found 3 mapping relations
FCLoopFindTopologyMappings: Final number of independent topologies: 2 \text{FCLoopFindTopologyMappings:
}\;\text{Final number of independent topologies: }2 FCLoopFindTopologyMappings: Final number of independent topologies: 2
( FCTopology ( fctopology3 , { 1 ( p3 2 + i η ) , 1 ( p2 2 + i η ) , 1 ( p1 2 + i η ) , 1 ( ( p2 + p3 ) 2 + i η ) , 1 ( ( p1 + p3 ) 2 + i η ) , 1 ( ( p2 − Q ) 2 + i η ) , 1 ( ( p2 + p3 − Q ) 2 + i η ) , 1 ( ( p1 + p3 − Q ) 2 + i η ) , 1 ( ( p1 + p2 + p3 − Q ) 2 + i η ) } , { p1 , p2 , p3 } , { Q } , { } , { } ) { p1 → − p1 − p3 + Q , p2 → − p2 − p3 + Q } G fctopology3 ( n1 _ , n7 _ , n8 _ , n5 _ , n6 _ , n4 _ , n2 _ , n3 _ , n9 _ ) : → G fctopology1 ( n1 , n2 , n3 , n4 , n5 , n6 , n7 , n8 , n9 ) FCTopology ( fctopology4 , { 1 ( p3 2 + i η ) , 1 ( p2 2 + i η ) , 1 ( p1 2 + i η ) , 1 ( ( p2 + p3 ) 2 + i η ) , 1 ( ( p1 + p3 ) 2 + i η ) , 1 ( ( p2 − Q ) 2 + i η ) , 1 ( ( p1 − Q ) 2 + i η ) , 1 ( ( p1 + p3 − Q ) 2 + i η ) , 1 ( ( p1 + p2 + p3 − Q ) 2 + i η ) } , { p1 , p2 , p3 } , { Q } , { } , { } ) { p1 → Q − p2 , p2 → Q − p1 , p3 → − p3 } G fctopology4 ( n1 _ , n6 _ , n5 _ , n8 _ , n7 _ , n3 _ , n2 _ , n4 _ , n9 _ ) : → G fctopology1 ( n1 , n2 , n3 , n4 , n5 , n6 , n7 , n8 , n9 ) FCTopology ( fctopology5 , { 1 ( p3 2 + i η ) , 1 ( p2 2 + i η ) , 1 ( p1 2 + i η ) , 1 ( ( p1 + p3 ) 2 + i η ) , 1 ( ( p2 − Q ) 2 + i η ) , 1 ( ( p1 − Q ) 2 + i η ) , 1 ( ( p1 + p3 − Q ) 2 + i η ) , 1 ( ( p1 + p2 − Q ) 2 + i η ) , 1 ( ( p1 + p2 + p3 − Q ) 2 + i η ) } , { p1 , p2 , p3 } , { Q } , { } , { } ) { p1 → p2 , p2 → p1 } G fctopology5 ( n1 _ , n3 _ , n2 _ , n4 _ , n6 _ , n5 _ , n7 _ , n8 _ , n9 _ ) : → G fctopology2 ( n1 , n2 , n3 , n4 , n5 , n6 , n7 , n8 , n9 ) ) \left(
\begin{array}{ccc}
\;\text{FCTopology}\left(\text{fctopology3},\left\{\frac{1}{(\text{p3}^2+i
\eta )},\frac{1}{(\text{p2}^2+i \eta )},\frac{1}{(\text{p1}^2+i \eta
)},\frac{1}{((\text{p2}+\text{p3})^2+i \eta
)},\frac{1}{((\text{p1}+\text{p3})^2+i \eta
)},\frac{1}{((\text{p2}-Q)^2+i \eta
)},\frac{1}{((\text{p2}+\text{p3}-Q)^2+i \eta
)},\frac{1}{((\text{p1}+\text{p3}-Q)^2+i \eta
)},\frac{1}{((\text{p1}+\text{p2}+\text{p3}-Q)^2+i \eta
)}\right\},\{\text{p1},\text{p2},\text{p3}\},\{Q\},\{\},\{\}\right)
& \{\text{p1}\to -\text{p1}-\text{p3}+Q,\text{p2}\to
-\text{p2}-\text{p3}+Q\} &
G^{\text{fctopology3}}(\text{n1$\_$},\text{n7$\_$},\text{n8$\_$},\text{n5$\_$},\text{n6$\_$},\text{n4$\_$},\text{n2$\_$},\text{n3$\_$},\text{n9$\_$}):\to
G^{\text{fctopology1}}(\text{n1},\text{n2},\text{n3},\text{n4},\text{n5},\text{n6},\text{n7},\text{n8},\text{n9})
\\
\;\text{FCTopology}\left(\text{fctopology4},\left\{\frac{1}{(\text{p3}^2+i
\eta )},\frac{1}{(\text{p2}^2+i \eta )},\frac{1}{(\text{p1}^2+i \eta
)},\frac{1}{((\text{p2}+\text{p3})^2+i \eta
)},\frac{1}{((\text{p1}+\text{p3})^2+i \eta
)},\frac{1}{((\text{p2}-Q)^2+i \eta )},\frac{1}{((\text{p1}-Q)^2+i \eta
)},\frac{1}{((\text{p1}+\text{p3}-Q)^2+i \eta
)},\frac{1}{((\text{p1}+\text{p2}+\text{p3}-Q)^2+i \eta
)}\right\},\{\text{p1},\text{p2},\text{p3}\},\{Q\},\{\},\{\}\right)
& \{\text{p1}\to Q-\text{p2},\text{p2}\to Q-\text{p1},\text{p3}\to
-\text{p3}\} &
G^{\text{fctopology4}}(\text{n1$\_$},\text{n6$\_$},\text{n5$\_$},\text{n8$\_$},\text{n7$\_$},\text{n3$\_$},\text{n2$\_$},\text{n4$\_$},\text{n9$\_$}):\to
G^{\text{fctopology1}}(\text{n1},\text{n2},\text{n3},\text{n4},\text{n5},\text{n6},\text{n7},\text{n8},\text{n9})
\\
\;\text{FCTopology}\left(\text{fctopology5},\left\{\frac{1}{(\text{p3}^2+i
\eta )},\frac{1}{(\text{p2}^2+i \eta )},\frac{1}{(\text{p1}^2+i \eta
)},\frac{1}{((\text{p1}+\text{p3})^2+i \eta
)},\frac{1}{((\text{p2}-Q)^2+i \eta )},\frac{1}{((\text{p1}-Q)^2+i \eta
)},\frac{1}{((\text{p1}+\text{p3}-Q)^2+i \eta
)},\frac{1}{((\text{p1}+\text{p2}-Q)^2+i \eta
)},\frac{1}{((\text{p1}+\text{p2}+\text{p3}-Q)^2+i \eta
)}\right\},\{\text{p1},\text{p2},\text{p3}\},\{Q\},\{\},\{\}\right)
& \{\text{p1}\to \;\text{p2},\text{p2}\to \;\text{p1}\} &
G^{\text{fctopology5}}(\text{n1$\_$},\text{n3$\_$},\text{n2$\_$},\text{n4$\_$},\text{n6$\_$},\text{n5$\_$},\text{n7$\_$},\text{n8$\_$},\text{n9$\_$}):\to
G^{\text{fctopology2}}(\text{n1},\text{n2},\text{n3},\text{n4},\text{n5},\text{n6},\text{n7},\text{n8},\text{n9})
\\
\end{array}
\right) FCTopology ( fctopology3 , { ( p3 2 + i η ) 1 , ( p2 2 + i η ) 1 , ( p1 2 + i η ) 1 , (( p2 + p3 ) 2 + i η ) 1 , (( p1 + p3 ) 2 + i η ) 1 , (( p2 − Q ) 2 + i η ) 1 , (( p2 + p3 − Q ) 2 + i η ) 1 , (( p1 + p3 − Q ) 2 + i η ) 1 , (( p1 + p2 + p3 − Q ) 2 + i η ) 1 } , { p1 , p2 , p3 } , { Q } , { } , { } ) FCTopology ( fctopology4 , { ( p3 2 + i η ) 1 , ( p2 2 + i η ) 1 , ( p1 2 + i η ) 1 , (( p2 + p3 ) 2 + i η ) 1 , (( p1 + p3 ) 2 + i η ) 1 , (( p2 − Q ) 2 + i η ) 1 , (( p1 − Q ) 2 + i η ) 1 , (( p1 + p3 − Q ) 2 + i η ) 1 , (( p1 + p2 + p3 − Q ) 2 + i η ) 1 } , { p1 , p2 , p3 } , { Q } , { } , { } ) FCTopology ( fctopology5 , { ( p3 2 + i η ) 1 , ( p2 2 + i η ) 1 , ( p1 2 + i η ) 1 , (( p1 + p3 ) 2 + i η ) 1 , (( p2 − Q ) 2 + i η ) 1 , (( p1 − Q ) 2 + i η ) 1 , (( p1 + p3 − Q ) 2 + i η ) 1 , (( p1 + p2 − Q ) 2 + i η ) 1 , (( p1 + p2 + p3 − Q ) 2 + i η ) 1 } , { p1 , p2 , p3 } , { Q } , { } , { } ) { p1 → − p1 − p3 + Q , p2 → − p2 − p3 + Q } { p1 → Q − p2 , p2 → Q − p1 , p3 → − p3 } { p1 → p2 , p2 → p1 } G fctopology3 ( n1_ , n7_ , n8_ , n5_ , n6_ , n4_ , n2_ , n3_ , n9_ ) :→ G fctopology1 ( n1 , n2 , n3 , n4 , n5 , n6 , n7 , n8 , n9 ) G fctopology4 ( n1_ , n6_ , n5_ , n8_ , n7_ , n3_ , n2_ , n4_ , n9_ ) :→ G fctopology1 ( n1 , n2 , n3 , n4 , n5 , n6 , n7 , n8 , n9 ) G fctopology5 ( n1_ , n3_ , n2_ , n4_ , n6_ , n5_ , n7_ , n8_ , n9_ ) :→ G fctopology2 ( n1 , n2 , n3 , n4 , n5 , n6 , n7 , n8 , n9 )
And these are the final topologies
{ FCTopology ( fctopology1 , { 1 ( p3 2 + i η ) , 1 ( p2 2 + i η ) , 1 ( p1 2 + i η ) , 1 ( ( p2 + p3 ) 2 + i η ) , 1 ( ( p2 − Q ) 2 + i η ) , 1 ( ( p1 − Q ) 2 + i η ) , 1 ( ( p2 + p3 − Q ) 2 + i η ) , 1 ( ( p1 + p3 − Q ) 2 + i η ) , 1 ( ( p1 + p2 + p3 − Q ) 2 + i η ) } , { p1 , p2 , p3 } , { Q } , { } , { } ) , FCTopology ( fctopology2 , { 1 ( p3 2 + i η ) , 1 ( p2 2 + i η ) , 1 ( p1 2 + i η ) , 1 ( ( p2 + p3 ) 2 + i η ) , 1 ( ( p2 − Q ) 2 + i η ) , 1 ( ( p1 − Q ) 2 + i η ) , 1 ( ( p2 + p3 − Q ) 2 + i η ) , 1 ( ( p1 + p2 − Q ) 2 + i η ) , 1 ( ( p1 + p2 + p3 − Q ) 2 + i η ) } , { p1 , p2 , p3 } , { Q } , { } , { } ) } \left\{\text{FCTopology}\left(\text{fctopology1},\left\{\frac{1}{(\text{p3}^2+i
\eta )},\frac{1}{(\text{p2}^2+i \eta )},\frac{1}{(\text{p1}^2+i \eta
)},\frac{1}{((\text{p2}+\text{p3})^2+i \eta
)},\frac{1}{((\text{p2}-Q)^2+i \eta )},\frac{1}{((\text{p1}-Q)^2+i \eta
)},\frac{1}{((\text{p2}+\text{p3}-Q)^2+i \eta
)},\frac{1}{((\text{p1}+\text{p3}-Q)^2+i \eta
)},\frac{1}{((\text{p1}+\text{p2}+\text{p3}-Q)^2+i \eta
)}\right\},\{\text{p1},\text{p2},\text{p3}\},\{Q\},\{\},\{\}\right),\text{FCTopology}\left(\text{fctopology2},\left\{\frac{1}{(\text{p3}^2+i
\eta )},\frac{1}{(\text{p2}^2+i \eta )},\frac{1}{(\text{p1}^2+i \eta
)},\frac{1}{((\text{p2}+\text{p3})^2+i \eta
)},\frac{1}{((\text{p2}-Q)^2+i \eta )},\frac{1}{((\text{p1}-Q)^2+i \eta
)},\frac{1}{((\text{p2}+\text{p3}-Q)^2+i \eta
)},\frac{1}{((\text{p1}+\text{p2}-Q)^2+i \eta
)},\frac{1}{((\text{p1}+\text{p2}+\text{p3}-Q)^2+i \eta
)}\right\},\{\text{p1},\text{p2},\text{p3}\},\{Q\},\{\},\{\}\right)\right\} { FCTopology ( fctopology1 , { ( p3 2 + i η ) 1 , ( p2 2 + i η ) 1 , ( p1 2 + i η ) 1 , (( p2 + p3 ) 2 + i η ) 1 , (( p2 − Q ) 2 + i η ) 1 , (( p1 − Q ) 2 + i η ) 1 , (( p2 + p3 − Q ) 2 + i η ) 1 , (( p1 + p3 − Q ) 2 + i η ) 1 , (( p1 + p2 + p3 − Q ) 2 + i η ) 1 } , { p1 , p2 , p3 } , { Q } , { } , { } ) , FCTopology ( fctopology2 , { ( p3 2 + i η ) 1 , ( p2 2 + i η ) 1 , ( p1 2 + i η ) 1 , (( p2 + p3 ) 2 + i η ) 1 , (( p2 − Q ) 2 + i η ) 1 , (( p1 − Q ) 2 + i η ) 1 , (( p2 + p3 − Q ) 2 + i η ) 1 , (( p1 + p2 − Q ) 2 + i η ) 1 , (( p1 + p2 + p3 − Q ) 2 + i η ) 1 } , { p1 , p2 , p3 } , { Q } , { } , { } ) }
Here is another example
= { FCTopology[ fctopology1, { SFAD[{{ q2, 0 }, { 0 , 1 }, 1 }], [{{ q1, 0 }, { 0 , 1 }, 1 }], SFAD[{{ q1 + q2, 0 }, { 0 , 1 }, 1 }], SFAD[{{ p + q1, 0 }, { 0 , 1 }, 1 }], [{{ p - q2, 0 }, { 0 , 1 }, 1 }]}, { q1, q2}, { p }, {}, {}], [ fctopology2, { SFAD[{{ q2, 0 }, { 0 , 1 }, 1 }], SFAD[{{ q1, 0 }, { 0 , 1 }, 1 }], [{{ p + q2, 0 }, { 0 , 1 }, 1 }], SFAD[{{ p - q1, 0 }, { 0 , 1 }, 1 }]}, { q1, q2}, { p }, {}, {}], [ fctopology3, { SFAD[{{ q2, 0 }, { 0 , 1 }, 1 }], SFAD[{{ q1, 0 }, { 0 , 1 }, 1 }], [{{ p - q1, 0 }, { 0 , 1 }, 1 }], SFAD[{{ p - q1 + q2, 0 }, { 0 , 1 }, 1 }]}, { q1, q2}, { p }, {}, {}]} { FCTopology ( fctopology1 , { 1 ( q2 2 + i η ) , 1 ( q1 2 + i η ) , 1 ( ( q1 + q2 ) 2 + i η ) , 1 ( ( p + q1 ) 2 + i η ) , 1 ( ( p − q2 ) 2 + i η ) } , { q1 , q2 } , { p } , { } , { } ) , FCTopology ( fctopology2 , { 1 ( q2 2 + i η ) , 1 ( q1 2 + i η ) , 1 ( ( p + q2 ) 2 + i η ) , 1 ( ( p − q1 ) 2 + i η ) } , { q1 , q2 } , { p } , { } , { } ) , FCTopology ( fctopology3 , { 1 ( q2 2 + i η ) , 1 ( q1 2 + i η ) , 1 ( ( p − q1 ) 2 + i η ) , 1 ( ( p − q1 + q2 ) 2 + i η ) } , { q1 , q2 } , { p } , { } , { } ) } \left\{\text{FCTopology}\left(\text{fctopology1},\left\{\frac{1}{(\text{q2}^2+i
\eta )},\frac{1}{(\text{q1}^2+i \eta
)},\frac{1}{((\text{q1}+\text{q2})^2+i \eta
)},\frac{1}{((p+\text{q1})^2+i \eta )},\frac{1}{((p-\text{q2})^2+i \eta
)}\right\},\{\text{q1},\text{q2}\},\{p\},\{\},\{\}\right),\text{FCTopology}\left(\text{fctopology2},\left\{\frac{1}{(\text{q2}^2+i
\eta )},\frac{1}{(\text{q1}^2+i \eta )},\frac{1}{((p+\text{q2})^2+i \eta
)},\frac{1}{((p-\text{q1})^2+i \eta
)}\right\},\{\text{q1},\text{q2}\},\{p\},\{\},\{\}\right),\text{FCTopology}\left(\text{fctopology3},\left\{\frac{1}{(\text{q2}^2+i
\eta )},\frac{1}{(\text{q1}^2+i \eta )},\frac{1}{((p-\text{q1})^2+i \eta
)},\frac{1}{((p-\text{q1}+\text{q2})^2+i \eta
)}\right\},\{\text{q1},\text{q2}\},\{p\},\{\},\{\}\right)\right\} { FCTopology ( fctopology1 , { ( q2 2 + i η ) 1 , ( q1 2 + i η ) 1 , (( q1 + q2 ) 2 + i η ) 1 , (( p + q1 ) 2 + i η ) 1 , (( p − q2 ) 2 + i η ) 1 } , { q1 , q2 } , { p } , { } , { } ) , FCTopology ( fctopology2 , { ( q2 2 + i η ) 1 , ( q1 2 + i η ) 1 , (( p + q2 ) 2 + i η ) 1 , (( p − q1 ) 2 + i η ) 1 } , { q1 , q2 } , { p } , { } , { } ) , FCTopology ( fctopology3 , { ( q2 2 + i η ) 1 , ( q1 2 + i η ) 1 , (( p − q1 ) 2 + i η ) 1 , (( p − q1 + q2 ) 2 + i η ) 1 } , { q1 , q2 } , { p } , { } , { } ) }
Yet this time we have some preferred set of topologies and want to
match to them (if possible)
= { FCTopology[ prop2L, { SFAD[{{ q1, 0 }, { 0 , 1 }, 1 }], [{{ q2, 0 }, { 0 , 1 }, 1 }], SFAD[{{ q1 - q2, 0 }, { 0 , 1 }, 1 }], SFAD[{{ - p + q1, 0 }, { 0 , 1 }, 1 }], [{{ - p + q2, 0 }, { 0 , 1 }, 1 }]}, { q1, q2}, { p }, {}, {}], [ prop2LX1, { SFAD[{{ q2, 0 }, { 0 , 1 }, 1 }], SFAD[{{ q1 - q2, 0 }, { 0 , 1 }, 1 }], [{{ - p + q1, 0 }, { 0 , 1 }, 1 }], SFAD[{{ - p + q2, 0 }, { 0 , 1 }, 1 }]}, { q1, q2}, { p }, {}, {}], [ prop2LX3, { SFAD[{{ q1, 0 }, { 0 , 1 }, 1 }], SFAD[{{ q2, 0 }, { 0 , 1 }, 1 }], [{{ - p + q1, 0 }, { 0 , 1 }, 1 }], SFAD[{{ - p + q2, 0 }, { 0 , 1 }, 1 }]}, { q1, q2}, { p }, {}, {}], [ prop2LX15, { SFAD[{{ q2, 0 }, { 0 , 1 }, 1 }], SFAD[{{ q1 - q2, 0 }, { 0 , 1 }, 1 }], [{{ - p + q1, 0 }, { 0 , 1 }, 1 }]}, { q1, q2}, { p }, {}, {}]} { FCTopology ( prop2L , { 1 ( q1 2 + i η ) , 1 ( q2 2 + i η ) , 1 ( ( q1 − q2 ) 2 + i η ) , 1 ( ( q1 − p ) 2 + i η ) , 1 ( ( q2 − p ) 2 + i η ) } , { q1 , q2 } , { p } , { } , { } ) , FCTopology ( prop2LX1 , { 1 ( q2 2 + i η ) , 1 ( ( q1 − q2 ) 2 + i η ) , 1 ( ( q1 − p ) 2 + i η ) , 1 ( ( q2 − p ) 2 + i η ) } , { q1 , q2 } , { p } , { } , { } ) , FCTopology ( prop2LX3 , { 1 ( q1 2 + i η ) , 1 ( q2 2 + i η ) , 1 ( ( q1 − p ) 2 + i η ) , 1 ( ( q2 − p ) 2 + i η ) } , { q1 , q2 } , { p } , { } , { } ) , FCTopology ( prop2LX15 , { 1 ( q2 2 + i η ) , 1 ( ( q1 − q2 ) 2 + i η ) , 1 ( ( q1 − p ) 2 + i η ) } , { q1 , q2 } , { p } , { } , { } ) } \left\{\text{FCTopology}\left(\text{prop2L},\left\{\frac{1}{(\text{q1}^2+i
\eta )},\frac{1}{(\text{q2}^2+i \eta
)},\frac{1}{((\text{q1}-\text{q2})^2+i \eta
)},\frac{1}{((\text{q1}-p)^2+i \eta )},\frac{1}{((\text{q2}-p)^2+i \eta
)}\right\},\{\text{q1},\text{q2}\},\{p\},\{\},\{\}\right),\text{FCTopology}\left(\text{prop2LX1},\left\{\frac{1}{(\text{q2}^2+i
\eta )},\frac{1}{((\text{q1}-\text{q2})^2+i \eta
)},\frac{1}{((\text{q1}-p)^2+i \eta )},\frac{1}{((\text{q2}-p)^2+i \eta
)}\right\},\{\text{q1},\text{q2}\},\{p\},\{\},\{\}\right),\text{FCTopology}\left(\text{prop2LX3},\left\{\frac{1}{(\text{q1}^2+i
\eta )},\frac{1}{(\text{q2}^2+i \eta )},\frac{1}{((\text{q1}-p)^2+i \eta
)},\frac{1}{((\text{q2}-p)^2+i \eta
)}\right\},\{\text{q1},\text{q2}\},\{p\},\{\},\{\}\right),\text{FCTopology}\left(\text{prop2LX15},\left\{\frac{1}{(\text{q2}^2+i
\eta )},\frac{1}{((\text{q1}-\text{q2})^2+i \eta
)},\frac{1}{((\text{q1}-p)^2+i \eta
)}\right\},\{\text{q1},\text{q2}\},\{p\},\{\},\{\}\right)\right\} { FCTopology ( prop2L , { ( q1 2 + i η ) 1 , ( q2 2 + i η ) 1 , (( q1 − q2 ) 2 + i η ) 1 , (( q1 − p ) 2 + i η ) 1 , (( q2 − p ) 2 + i η ) 1 } , { q1 , q2 } , { p } , { } , { } ) , FCTopology ( prop2LX1 , { ( q2 2 + i η ) 1 , (( q1 − q2 ) 2 + i η ) 1 , (( q1 − p ) 2 + i η ) 1 , (( q2 − p ) 2 + i η ) 1 } , { q1 , q2 } , { p } , { } , { } ) , FCTopology ( prop2LX3 , { ( q1 2 + i η ) 1 , ( q2 2 + i η ) 1 , (( q1 − p ) 2 + i η ) 1 , (( q2 − p ) 2 + i η ) 1 } , { q1 , q2 } , { p } , { } , { } ) , FCTopology ( prop2LX15 , { ( q2 2 + i η ) 1 , (( q1 − q2 ) 2 + i η ) 1 , (( q1 − p ) 2 + i η ) 1 } , { q1 , q2 } , { p } , { } , { } ) }
= FCLoopFindTopologyMappings[ topos2, PreferredTopologies -> preferredTopos2] ;FCLoopFindTopologyMappings: Found 3 mapping relations \text{FCLoopFindTopologyMappings:
}\;\text{Found }3\text{ mapping relations } FCLoopFindTopologyMappings: Found 3 mapping relations
FCLoopFindTopologyMappings: Final number of independent topologies: 3 \text{FCLoopFindTopologyMappings:
}\;\text{Final number of independent topologies: }3 FCLoopFindTopologyMappings: Final number of independent topologies: 3
( FCTopology ( fctopology1 , { 1 ( q2 2 + i η ) , 1 ( q1 2 + i η ) , 1 ( ( q1 + q2 ) 2 + i η ) , 1 ( ( p + q1 ) 2 + i η ) , 1 ( ( p − q2 ) 2 + i η ) } , { q1 , q2 } , { p } , { } , { } ) { q1 → − q2 , q2 → q1 } G fctopology1 ( n1 _ , n2 _ , n3 _ , n5 _ , n4 _ ) : → G prop2L ( n1 , n2 , n3 , n4 , n5 ) FCTopology ( fctopology2 , { 1 ( q2 2 + i η ) , 1 ( q1 2 + i η ) , 1 ( ( p + q2 ) 2 + i η ) , 1 ( ( p − q1 ) 2 + i η ) } , { q1 , q2 } , { p } , { } , { } ) { q1 → q2 , q2 → − q1 } G fctopology2 ( n1 _ , n2 _ , n3 _ , n4 _ ) : → G prop2LX3 ( n1 , n2 , n3 , n4 ) FCTopology ( fctopology3 , { 1 ( q2 2 + i η ) , 1 ( q1 2 + i η ) , 1 ( ( p − q1 ) 2 + i η ) , 1 ( ( p − q1 + q2 ) 2 + i η ) } , { q1 , q2 } , { p } , { } , { } ) { q1 → q2 , q2 → q2 − q1 } G fctopology3 ( n2 _ , n1 _ , n4 _ , n3 _ ) : → G prop2LX1 ( n1 , n2 , n3 , n4 ) ) \left(
\begin{array}{ccc}
\;\text{FCTopology}\left(\text{fctopology1},\left\{\frac{1}{(\text{q2}^2+i
\eta )},\frac{1}{(\text{q1}^2+i \eta
)},\frac{1}{((\text{q1}+\text{q2})^2+i \eta
)},\frac{1}{((p+\text{q1})^2+i \eta )},\frac{1}{((p-\text{q2})^2+i \eta
)}\right\},\{\text{q1},\text{q2}\},\{p\},\{\},\{\}\right) &
\{\text{q1}\to -\text{q2},\text{q2}\to \;\text{q1}\} &
G^{\text{fctopology1}}(\text{n1$\_$},\text{n2$\_$},\text{n3$\_$},\text{n5$\_$},\text{n4$\_$}):\to
G^{\text{prop2L}}(\text{n1},\text{n2},\text{n3},\text{n4},\text{n5}) \\
\;\text{FCTopology}\left(\text{fctopology2},\left\{\frac{1}{(\text{q2}^2+i
\eta )},\frac{1}{(\text{q1}^2+i \eta )},\frac{1}{((p+\text{q2})^2+i \eta
)},\frac{1}{((p-\text{q1})^2+i \eta
)}\right\},\{\text{q1},\text{q2}\},\{p\},\{\},\{\}\right) &
\{\text{q1}\to \;\text{q2},\text{q2}\to -\text{q1}\} &
G^{\text{fctopology2}}(\text{n1$\_$},\text{n2$\_$},\text{n3$\_$},\text{n4$\_$}):\to
G^{\text{prop2LX3}}(\text{n1},\text{n2},\text{n3},\text{n4}) \\
\;\text{FCTopology}\left(\text{fctopology3},\left\{\frac{1}{(\text{q2}^2+i
\eta )},\frac{1}{(\text{q1}^2+i \eta )},\frac{1}{((p-\text{q1})^2+i \eta
)},\frac{1}{((p-\text{q1}+\text{q2})^2+i \eta
)}\right\},\{\text{q1},\text{q2}\},\{p\},\{\},\{\}\right) &
\{\text{q1}\to \;\text{q2},\text{q2}\to \;\text{q2}-\text{q1}\} &
G^{\text{fctopology3}}(\text{n2$\_$},\text{n1$\_$},\text{n4$\_$},\text{n3$\_$}):\to
G^{\text{prop2LX1}}(\text{n1},\text{n2},\text{n3},\text{n4}) \\
\end{array}
\right) FCTopology ( fctopology1 , { ( q2 2 + i η ) 1 , ( q1 2 + i η ) 1 , (( q1 + q2 ) 2 + i η ) 1 , (( p + q1 ) 2 + i η ) 1 , (( p − q2 ) 2 + i η ) 1 } , { q1 , q2 } , { p } , { } , { } ) FCTopology ( fctopology2 , { ( q2 2 + i η ) 1 , ( q1 2 + i η ) 1 , (( p + q2 ) 2 + i η ) 1 , (( p − q1 ) 2 + i η ) 1 } , { q1 , q2 } , { p } , { } , { } ) FCTopology ( fctopology3 , { ( q2 2 + i η ) 1 , ( q1 2 + i η ) 1 , (( p − q1 ) 2 + i η ) 1 , (( p − q1 + q2 ) 2 + i η ) 1 } , { q1 , q2 } , { p } , { } , { } ) { q1 → − q2 , q2 → q1 } { q1 → q2 , q2 → − q1 } { q1 → q2 , q2 → q2 − q1 } G fctopology1 ( n1_ , n2_ , n3_ , n5_ , n4_ ) :→ G prop2L ( n1 , n2 , n3 , n4 , n5 ) G fctopology2 ( n1_ , n2_ , n3_ , n4_ ) :→ G prop2LX3 ( n1 , n2 , n3 , n4 ) G fctopology3 ( n2_ , n1_ , n4_ , n3_ ) :→ G prop2LX1 ( n1 , n2 , n3 , n4 )
And these are the final occurring topologies
{ FCTopology ( prop2L , { 1 ( q1 2 + i η ) , 1 ( q2 2 + i η ) , 1 ( ( q1 − q2 ) 2 + i η ) , 1 ( ( q1 − p ) 2 + i η ) , 1 ( ( q2 − p ) 2 + i η ) } , { q1 , q2 } , { p } , { } , { } ) , FCTopology ( prop2LX1 , { 1 ( q2 2 + i η ) , 1 ( ( q1 − q2 ) 2 + i η ) , 1 ( ( q1 − p ) 2 + i η ) , 1 ( ( q2 − p ) 2 + i η ) } , { q1 , q2 } , { p } , { } , { } ) , FCTopology ( prop2LX3 , { 1 ( q1 2 + i η ) , 1 ( q2 2 + i η ) , 1 ( ( q1 − p ) 2 + i η ) , 1 ( ( q2 − p ) 2 + i η ) } , { q1 , q2 } , { p } , { } , { } ) } \left\{\text{FCTopology}\left(\text{prop2L},\left\{\frac{1}{(\text{q1}^2+i
\eta )},\frac{1}{(\text{q2}^2+i \eta
)},\frac{1}{((\text{q1}-\text{q2})^2+i \eta
)},\frac{1}{((\text{q1}-p)^2+i \eta )},\frac{1}{((\text{q2}-p)^2+i \eta
)}\right\},\{\text{q1},\text{q2}\},\{p\},\{\},\{\}\right),\text{FCTopology}\left(\text{prop2LX1},\left\{\frac{1}{(\text{q2}^2+i
\eta )},\frac{1}{((\text{q1}-\text{q2})^2+i \eta
)},\frac{1}{((\text{q1}-p)^2+i \eta )},\frac{1}{((\text{q2}-p)^2+i \eta
)}\right\},\{\text{q1},\text{q2}\},\{p\},\{\},\{\}\right),\text{FCTopology}\left(\text{prop2LX3},\left\{\frac{1}{(\text{q1}^2+i
\eta )},\frac{1}{(\text{q2}^2+i \eta )},\frac{1}{((\text{q1}-p)^2+i \eta
)},\frac{1}{((\text{q2}-p)^2+i \eta
)}\right\},\{\text{q1},\text{q2}\},\{p\},\{\},\{\}\right)\right\} { FCTopology ( prop2L , { ( q1 2 + i η ) 1 , ( q2 2 + i η ) 1 , (( q1 − q2 ) 2 + i η ) 1 , (( q1 − p ) 2 + i η ) 1 , (( q2 − p ) 2 + i η ) 1 } , { q1 , q2 } , { p } , { } , { } ) , FCTopology ( prop2LX1 , { ( q2 2 + i η ) 1 , (( q1 − q2 ) 2 + i η ) 1 , (( q1 − p ) 2 + i η ) 1 , (( q2 − p ) 2 + i η ) 1 } , { q1 , q2 } , { p } , { } , { } ) , FCTopology ( prop2LX3 , { ( q1 2 + i η ) 1 , ( q2 2 + i η ) 1 , (( q1 − p ) 2 + i η ) 1 , (( q2 − p ) 2 + i η ) 1 } , { q1 , q2 } , { p } , { } , { } ) }
If we need to match subtopologies into larger topologies, we first
need to generate all possible subtopologies for each relevant
topology.
= { [ fctopology1, { [{{ l1 + l2 - q1, 0 }, { 0 , 1 }, 1 }], [{{ l2, 0 }, { SMP[ "m_t" ] ^ 2 , 1 }, 1 }], [{{ l1, 0 }, { SMP[ "m_t" ] ^ 2 , 1 }, 1 }], [{{ l2 + q2, 0 }, { SMP[ "m_t" ] ^ 2 , 1 }, 1 }], [{{ l1 - q1, 0 }, { SMP[ "m_t" ] ^ 2 , 1 }, 1 }], [{{ l1 - q1 - q2, 0 }, { SMP[ "m_t" ] ^ 2 , 1 }, 1 }]}, { l1, l2}, { q1, q2}, {}, {}], [ fctopology9, { [{{ l1 + l2 + q2, 0 }, { 0 , 1 }, 1 }], [{{ l2, 0 }, { SMP[ "m_t" ] ^ 2 , 1 }, 1 }], [{{ l1, 0 }, { SMP[ "m_t" ] ^ 2 , 1 }, 1 }], [{{ l1 + q2, 0 }, { SMP[ "m_t" ] ^ 2 , 1 }, 1 }], [{{ l1 - q1, 0 }, { SMP[ "m_t" ] ^ 2 , 1 }, 1 }]}, { l1, l2}, { q1, q2}, {}, {}] } { FCTopology ( fctopology1 , { 1 ( ( l1 + l2 − q1 ) 2 + i η ) , 1 ( l2 2 − m t 2 + i η ) , 1 ( l1 2 − m t 2 + i η ) , 1 ( ( l2 + q2 ) 2 − m t 2 + i η ) , 1 ( ( l1 − q1 ) 2 − m t 2 + i η ) , 1 ( ( l1 − q1 − q2 ) 2 − m t 2 + i η ) } , { l1 , l2 } , { q1 , q2 } , { } , { } ) , FCTopology ( fctopology9 , { 1 ( ( l1 + l2 + q2 ) 2 + i η ) , 1 ( l2 2 − m t 2 + i η ) , 1 ( l1 2 − m t 2 + i η ) , 1 ( ( l1 + q2 ) 2 − m t 2 + i η ) , 1 ( ( l1 − q1 ) 2 − m t 2 + i η ) } , { l1 , l2 } , { q1 , q2 } , { } , { } ) } \left\{\text{FCTopology}\left(\text{fctopology1},\left\{\frac{1}{((\text{l1}+\text{l2}-\text{q1})^2+i
\eta )},\frac{1}{(\text{l2}^2-m_t^2+i \eta
)},\frac{1}{(\text{l1}^2-m_t^2+i \eta
)},\frac{1}{((\text{l2}+\text{q2})^2-m_t^2+i \eta
)},\frac{1}{((\text{l1}-\text{q1})^2-m_t^2+i \eta
)},\frac{1}{((\text{l1}-\text{q1}-\text{q2})^2-m_t^2+i \eta
)}\right\},\{\text{l1},\text{l2}\},\{\text{q1},\text{q2}\},\{\},\{\}\right),\text{FCTopology}\left(\text{fctopology9},\left\{\frac{1}{((\text{l1}+\text{l2}+\text{q2})^2+i
\eta )},\frac{1}{(\text{l2}^2-m_t^2+i \eta
)},\frac{1}{(\text{l1}^2-m_t^2+i \eta
)},\frac{1}{((\text{l1}+\text{q2})^2-m_t^2+i \eta
)},\frac{1}{((\text{l1}-\text{q1})^2-m_t^2+i \eta
)}\right\},\{\text{l1},\text{l2}\},\{\text{q1},\text{q2}\},\{\},\{\}\right)\right\} { FCTopology ( fctopology1 , { (( l1 + l2 − q1 ) 2 + i η ) 1 , ( l2 2 − m t 2 + i η ) 1 , ( l1 2 − m t 2 + i η ) 1 , (( l2 + q2 ) 2 − m t 2 + i η ) 1 , (( l1 − q1 ) 2 − m t 2 + i η ) 1 , (( l1 − q1 − q2 ) 2 − m t 2 + i η ) 1 } , { l1 , l2 } , { q1 , q2 } , { } , { } ) , FCTopology ( fctopology9 , { (( l1 + l2 + q2 ) 2 + i η ) 1 , ( l2 2 − m t 2 + i η ) 1 , ( l1 2 − m t 2 + i η ) 1 , (( l1 + q2 ) 2 − m t 2 + i η ) 1 , (( l1 − q1 ) 2 − m t 2 + i η ) 1 } , { l1 , l2 } , { q1 , q2 } , { } , { } ) }
= Flatten [ FCLoopFindSubtopologies[ topos3]] ;37 37 37
Now we can match a smaller topology into a larger topology
= FCLoopFindTopologyMappings[ topos3, PreferredTopologies -> subTopos3] ;FCLoopFindTopologyMappings: Found 1 mapping relations \text{FCLoopFindTopologyMappings:
}\;\text{Found }1\text{ mapping relations } FCLoopFindTopologyMappings: Found 1 mapping relations
FCLoopFindTopologyMappings: Final number of independent topologies: 1 \text{FCLoopFindTopologyMappings:
}\;\text{Final number of independent topologies: }1 FCLoopFindTopologyMappings: Final number of independent topologies: 1
( FCTopology ( fctopology9 , { 1 ( ( l1 + l2 + q2 ) 2 + i η ) , 1 ( l2 2 − m t 2 + i η ) , 1 ( l1 2 − m t 2 + i η ) , 1 ( ( l1 + q2 ) 2 − m t 2 + i η ) , 1 ( ( l1 − q1 ) 2 − m t 2 + i η ) } , { l1 , l2 } , { q1 , q2 } , { } , { } ) { l1 → q1 − l1 , l2 → − l2 − q2 } G fctopology9 ( n1 _ , n3 _ , n4 _ , n5 _ , n2 _ ) : → G fctopology1 ( n1 , 0 , n2 , n3 , n4 , n5 ) ) \left(
\begin{array}{ccc}
\;\text{FCTopology}\left(\text{fctopology9},\left\{\frac{1}{((\text{l1}+\text{l2}+\text{q2})^2+i
\eta )},\frac{1}{(\text{l2}^2-m_t^2+i \eta
)},\frac{1}{(\text{l1}^2-m_t^2+i \eta
)},\frac{1}{((\text{l1}+\text{q2})^2-m_t^2+i \eta
)},\frac{1}{((\text{l1}-\text{q1})^2-m_t^2+i \eta
)}\right\},\{\text{l1},\text{l2}\},\{\text{q1},\text{q2}\},\{\},\{\}\right)
& \{\text{l1}\to \;\text{q1}-\text{l1},\text{l2}\to
-\text{l2}-\text{q2}\} &
G^{\text{fctopology9}}(\text{n1$\_$},\text{n3$\_$},\text{n4$\_$},\text{n5$\_$},\text{n2$\_$}):\to
G^{\text{fctopology1}}(\text{n1},0,\text{n2},\text{n3},\text{n4},\text{n5})
\\
\end{array}
\right) ( FCTopology ( fctopology9 , { (( l1 + l2 + q2 ) 2 + i η ) 1 , ( l2 2 − m t 2 + i η ) 1 , ( l1 2 − m t 2 + i η ) 1 , (( l1 + q2 ) 2 − m t 2 + i η ) 1 , (( l1 − q1 ) 2 − m t 2 + i η ) 1 } , { l1 , l2 } , { q1 , q2 } , { } , { } ) { l1 → q1 − l1 , l2 → − l2 − q2 } G fctopology9 ( n1_ , n3_ , n4_ , n5_ , n2_ ) :→ G fctopology1 ( n1 , 0 , n2 , n3 , n4 , n5 ) )
{ FCTopology ( fctopology1 , { 1 ( ( l1 + l2 − q1 ) 2 + i η ) , 1 ( l2 2 − m t 2 + i η ) , 1 ( l1 2 − m t 2 + i η ) , 1 ( ( l2 + q2 ) 2 − m t 2 + i η ) , 1 ( ( l1 − q1 ) 2 − m t 2 + i η ) , 1 ( ( l1 − q1 − q2 ) 2 − m t 2 + i η ) } , { l1 , l2 } , { q1 , q2 } , { } , { } ) } \left\{\text{FCTopology}\left(\text{fctopology1},\left\{\frac{1}{((\text{l1}+\text{l2}-\text{q1})^2+i
\eta )},\frac{1}{(\text{l2}^2-m_t^2+i \eta
)},\frac{1}{(\text{l1}^2-m_t^2+i \eta
)},\frac{1}{((\text{l2}+\text{q2})^2-m_t^2+i \eta
)},\frac{1}{((\text{l1}-\text{q1})^2-m_t^2+i \eta
)},\frac{1}{((\text{l1}-\text{q1}-\text{q2})^2-m_t^2+i \eta
)}\right\},\{\text{l1},\text{l2}\},\{\text{q1},\text{q2}\},\{\},\{\}\right)\right\} { FCTopology ( fctopology1 , { (( l1 + l2 − q1 ) 2 + i η ) 1 , ( l2 2 − m t 2 + i η ) 1 , ( l1 2 − m t 2 + i η ) 1 , (( l2 + q2 ) 2 − m t 2 + i η ) 1 , (( l1 − q1 ) 2 − m t 2 + i η ) 1 , (( l1 − q1 − q2 ) 2 − m t 2 + i η ) 1 } , { l1 , l2 } , { q1 , q2 } , { } , { } ) }
Mapping the following two topologies onto each other requires shifts
in the external momenta due to the chosen kinematic constraints.
= { [ topo1, { [{{ l1 + q1, 0 }, { m ^ 2 , 1 }, 1 }], [{{ l1 - l2, 0 }, { 0 , 1 }, 1 }], [{{ l2 + q1, 0 }, { m ^ 2 , 1 }, 1 }], [{{ l2 - q2, 0 }, { m ^ 2 , 1 }, 1 }], [{{ l2, 0 }, { 0 , 1 }, 1 }]}, { l1, l2}, { q1, q2}, { SPD[ q1, q1] -> 0 , SPD[ q2, q2] -> 0 , SPD[ q1, q2] -> s / 2 }, {}], [ topo2, { [{{ l1 - l2, 0 }, { m ^ 2 , 1 }, 1 }], [{{ l1 - q2, 0 }, { 0 , 1 }, 1 }], [{{ l2 - q2, 0 }, { m ^ 2 , 1 }, 1 }], [{{ l2 + q1, 0 }, { m ^ 2 , 1 }, 1 }], [{{ l2, 0 }, { 0 , 1 }, 1 }]}, { l1, l2}, { q1, q2}, { SPD[ q1, q1] -> 0 , SPD[ q2, q2] -> 0 , SPD[ q1, q2] -> s / 2 }, {}]} { FCTopology ( topo1 , { 1 ( ( l1 + q1 ) 2 − m 2 + i η ) , 1 ( ( l1 − l2 ) 2 + i η ) , 1 ( ( l2 + q1 ) 2 − m 2 + i η ) , 1 ( ( l2 − q2 ) 2 − m 2 + i η ) , 1 ( l2 2 + i η ) } , { l1 , l2 } , { q1 , q2 } , { q1 2 → 0 , q2 2 → 0 , q1 ⋅ q2 → s 2 } , { } ) , FCTopology ( topo2 , { 1 ( ( l1 − l2 ) 2 − m 2 + i η ) , 1 ( ( l1 − q2 ) 2 + i η ) , 1 ( ( l2 − q2 ) 2 − m 2 + i η ) , 1 ( ( l2 + q1 ) 2 − m 2 + i η ) , 1 ( l2 2 + i η ) } , { l1 , l2 } , { q1 , q2 } , { q1 2 → 0 , q2 2 → 0 , q1 ⋅ q2 → s 2 } , { } ) } \left\{\text{FCTopology}\left(\text{topo1},\left\{\frac{1}{((\text{l1}+\text{q1})^2-m^2+i
\eta )},\frac{1}{((\text{l1}-\text{l2})^2+i \eta
)},\frac{1}{((\text{l2}+\text{q1})^2-m^2+i \eta
)},\frac{1}{((\text{l2}-\text{q2})^2-m^2+i \eta
)},\frac{1}{(\text{l2}^2+i \eta
)}\right\},\{\text{l1},\text{l2}\},\{\text{q1},\text{q2}\},\left\{\text{q1}^2\to
0,\text{q2}^2\to 0,\text{q1}\cdot \;\text{q2}\to
\frac{s}{2}\right\},\{\}\right),\text{FCTopology}\left(\text{topo2},\left\{\frac{1}{((\text{l1}-\text{l2})^2-m^2+i
\eta )},\frac{1}{((\text{l1}-\text{q2})^2+i \eta
)},\frac{1}{((\text{l2}-\text{q2})^2-m^2+i \eta
)},\frac{1}{((\text{l2}+\text{q1})^2-m^2+i \eta
)},\frac{1}{(\text{l2}^2+i \eta
)}\right\},\{\text{l1},\text{l2}\},\{\text{q1},\text{q2}\},\left\{\text{q1}^2\to
0,\text{q2}^2\to 0,\text{q1}\cdot \;\text{q2}\to
\frac{s}{2}\right\},\{\}\right)\right\} { FCTopology ( topo1 , { (( l1 + q1 ) 2 − m 2 + i η ) 1 , (( l1 − l2 ) 2 + i η ) 1 , (( l2 + q1 ) 2 − m 2 + i η ) 1 , (( l2 − q2 ) 2 − m 2 + i η ) 1 , ( l2 2 + i η ) 1 } , { l1 , l2 } , { q1 , q2 } , { q1 2 → 0 , q2 2 → 0 , q1 ⋅ q2 → 2 s } , { } ) , FCTopology ( topo2 , { (( l1 − l2 ) 2 − m 2 + i η ) 1 , (( l1 − q2 ) 2 + i η ) 1 , (( l2 − q2 ) 2 − m 2 + i η ) 1 , (( l2 + q1 ) 2 − m 2 + i η ) 1 , ( l2 2 + i η ) 1 } , { l1 , l2 } , { q1 , q2 } , { q1 2 → 0 , q2 2 → 0 , q1 ⋅ q2 → 2 s } , { } ) }
= FCLoopFindTopologyMappings[ topos4, Momentum -> All ] ;FCLoopFindTopologyMappings: Found 1 mapping relations \text{FCLoopFindTopologyMappings:
}\;\text{Found }1\text{ mapping relations } FCLoopFindTopologyMappings: Found 1 mapping relations
FCLoopFindTopologyMappings: Final number of independent topologies: 1 \text{FCLoopFindTopologyMappings:
}\;\text{Final number of independent topologies: }1 FCLoopFindTopologyMappings: Final number of independent topologies: 1
( FCTopology ( topo2 , { 1 ( ( l1 − l2 ) 2 − m 2 + i η ) , 1 ( ( l1 − q2 ) 2 + i η ) , 1 ( ( l2 − q2 ) 2 − m 2 + i η ) , 1 ( ( l2 + q1 ) 2 − m 2 + i η ) , 1 ( l2 2 + i η ) } , { l1 , l2 } , { q1 , q2 } , { q1 2 → 0 , q2 2 → 0 , q1 ⋅ q2 → s 2 } , { } ) { l1 → − l1 + l2 − q1 , q1 → − q2 , q2 → − q1 } G topo2 ( n1 _ , n2 _ , n3 _ , n4 _ , n5 _ ) : → G topo1 ( n1 , n2 , n3 , n4 , n5 ) ) \left(
\begin{array}{ccc}
\;\text{FCTopology}\left(\text{topo2},\left\{\frac{1}{((\text{l1}-\text{l2})^2-m^2+i
\eta )},\frac{1}{((\text{l1}-\text{q2})^2+i \eta
)},\frac{1}{((\text{l2}-\text{q2})^2-m^2+i \eta
)},\frac{1}{((\text{l2}+\text{q1})^2-m^2+i \eta
)},\frac{1}{(\text{l2}^2+i \eta
)}\right\},\{\text{l1},\text{l2}\},\{\text{q1},\text{q2}\},\left\{\text{q1}^2\to
0,\text{q2}^2\to 0,\text{q1}\cdot \;\text{q2}\to
\frac{s}{2}\right\},\{\}\right) & \{\text{l1}\to
-\text{l1}+\text{l2}-\text{q1},\text{q1}\to -\text{q2},\text{q2}\to
-\text{q1}\} &
G^{\text{topo2}}(\text{n1$\_$},\text{n2$\_$},\text{n3$\_$},\text{n4$\_$},\text{n5$\_$}):\to
G^{\text{topo1}}(\text{n1},\text{n2},\text{n3},\text{n4},\text{n5}) \\
\end{array}
\right) ( FCTopology ( topo2 , { (( l1 − l2 ) 2 − m 2 + i η ) 1 , (( l1 − q2 ) 2 + i η ) 1 , (( l2 − q2 ) 2 − m 2 + i η ) 1 , (( l2 + q1 ) 2 − m 2 + i η ) 1 , ( l2 2 + i η ) 1 } , { l1 , l2 } , { q1 , q2 } , { q1 2 → 0 , q2 2 → 0 , q1 ⋅ q2 → 2 s } , { } ) { l1 → − l1 + l2 − q1 , q1 → − q2 , q2 → − q1 } G topo2 ( n1_ , n2_ , n3_ , n4_ , n5_ ) :→ G topo1 ( n1 , n2 , n3 , n4 , n5 ) )
Otherwise no mappings exist
[ topos4][[ 1 ]] FCLoopFindMomentumShifts: Failed to derive the momentum shifts between topologies topo2 and topo1. This can be due to the presence of nonquadratic propagators or because shifts in external momenta are also necessary. \text{FCLoopFindMomentumShifts:
}\;\text{Failed to derive the momentum shifts between topologies topo2
and topo1. This can be due to the presence of nonquadratic propagators
or because shifts in external momenta are also necessary.} FCLoopFindMomentumShifts: Failed to derive the momentum shifts between topologies topo2 and topo1. This can be due to the presence of nonquadratic propagators or because shifts in external momenta are also necessary.
FCLoopFindTopologyMappings: Found 0 mapping relations \text{FCLoopFindTopologyMappings:
}\;\text{Found }0\text{ mapping relations } FCLoopFindTopologyMappings: Found 0 mapping relations
FCLoopFindTopologyMappings: Final number of independent topologies: 2 \text{FCLoopFindTopologyMappings:
}\;\text{Final number of independent topologies: }2 FCLoopFindTopologyMappings: Final number of independent topologies: 2
{ } \{\} { }
Topologies containing eikonal or other nonstandard propagators may
introduce additional challenges. Even though two such topologies can be
recognized to be identical, the code still would not be able to work out
the correct momentum shifts without some additional input.
= FCTopology[ mytopo67, { SFAD[{{ k2, 0 }, { 0 , 1 }, 1 }], SFAD[{{ k1, 0 }, { 0 , 1 }, 1 }], [{{ k1 + k2, 0 }, { 0 , 1 }, 1 }], SFAD[{{ 0 , - k1 . nb}, { 0 , 1 }, 1 }], [{{ k2, - meta u0b k2 . nb}, { 0 , 1 }, 1 }], SFAD[{{ k1 + k2, - 2 gkin meta u0b (k1 + k2) . n }, { 0 , 1 }, 1 }], SFAD[{{ k1, - 2 gkin meta k1 . n + meta u0b k1 . nb}, { 2 gkin meta^ 2 u0b, 1 }, 1 }]}, { k1, k2}, { n , nb}, { Hold [ SPD][ n ] -> 0 , Hold [ SPD][ nb] -> 0 , Hold [ SPD][ n , nb] -> 2 }, {}] ;= FCTopology[ mytopo79, { SFAD[{{ k2, 0 }, { 0 , 1 }, 1 }], SFAD[{{ k1, 0 }, { 0 , 1 }, 1 }], [{{ 0 , k1 . nb}, { 0 , 1 }, 1 }], SFAD[{{ k2, - meta u0b k2 . nb}, { 0 , 1 }, 1 }], [{{ k1 + k2, - meta u0b (k1 + k2) . nb}, { 0 , 1 }, 1 }], SFAD[{{ k1, 2 gkin meta k1 . n - meta u0b k1 . nb}, { 2 gkin meta^ 2 u0b, 1 }, 1 }], [{{ k1 + k2, 2 gkin meta u0b (k1 + k2) . n - meta u0b (k1 + k2) . nb}, { 2 gkin meta^ 2 u0b^ 2 , 1 }, 1 }]}, { k1, k2}, { n , nb}, { Hold [ SPD][ n ] -> 0 , Hold [ SPD][ nb] -> 0 , Hold [ SPD][ n , nb] -> 2 }, {}] ;[ gkin, FCVariable] = True ;[ meta, FCVariable] = True ;[ u0b, FCVariable] = True ;At first sight these two topologies are independent from each
other
[{ topoEik1, topoEik2}] ;FCLoopFindMomentumShifts: The topologies contain following mixed quadratic-eikonal propagators that complicate the determination of the shifts: { 1 ( k1 2 + k1 ⋅ ( 2 gkin meta n − meta u0b nb ) − 2 gkin meta 2 u0b + i η ) , 1 ( k1 2 + k1 ⋅ ( meta u0b nb − 2 gkin meta n ) − 2 gkin meta 2 u0b + i η ) , 1 ( k2 2 − meta u0b ( k2 ⋅ nb ) + i η ) , 1 ( ( k1 + k2 ) 2 − 2 gkin meta u0b ( ( k1 + k2 ) ⋅ n ) + i η ) , 1 ( ( k1 + k2 ) 2 − meta u0b ( ( k1 + k2 ) ⋅ nb ) + i η ) , 1 ( ( k1 + k2 ) 2 + ( k1 + k2 ) ⋅ ( 2 gkin meta u0b n − meta u0b nb ) − 2 gkin meta 2 u0b 2 + i η ) } \text{FCLoopFindMomentumShifts:
}\;\text{The topologies contain following mixed quadratic-eikonal
propagators that complicate the determination of the shifts:
}\left\{\frac{1}{(\text{k1}^2+\text{k1}\cdot (2 \;\text{gkin}
\;\text{meta} n-\text{meta} \;\text{u0b} \;\text{nb})-2 \;\text{gkin}
\;\text{meta}^2 \;\text{u0b}+i \eta
)},\frac{1}{(\text{k1}^2+\text{k1}\cdot (\text{meta} \;\text{u0b}
\;\text{nb}-2 \;\text{gkin} \;\text{meta} n)-2 \;\text{gkin}
\;\text{meta}^2 \;\text{u0b}+i \eta )},\frac{1}{(\text{k2}^2-\text{meta}
\;\text{u0b} (\text{k2}\cdot \;\text{nb})+i \eta
)},\frac{1}{((\text{k1}+\text{k2})^2-2 \;\text{gkin} \;\text{meta}
\;\text{u0b} ((\text{k1}+\text{k2})\cdot n)+i \eta
)},\frac{1}{((\text{k1}+\text{k2})^2-\text{meta} \;\text{u0b}
((\text{k1}+\text{k2})\cdot \;\text{nb})+i \eta
)},\frac{1}{((\text{k1}+\text{k2})^2+(\text{k1}+\text{k2})\cdot (2
\;\text{gkin} \;\text{meta} \;\text{u0b} n-\text{meta} \;\text{u0b}
\;\text{nb})-2 \;\text{gkin} \;\text{meta}^2 \;\text{u0b}^2+i \eta
)}\right\} FCLoopFindMomentumShifts: The topologies contain following mixed quadratic-eikonal propagators that complicate the determination of the shifts: { ( k1 2 + k1 ⋅ ( 2 gkin meta n − meta u0b nb ) − 2 gkin meta 2 u0b + i η ) 1 , ( k1 2 + k1 ⋅ ( meta u0b nb − 2 gkin meta n ) − 2 gkin meta 2 u0b + i η ) 1 , ( k2 2 − meta u0b ( k2 ⋅ nb ) + i η ) 1 , (( k1 + k2 ) 2 − 2 gkin meta u0b (( k1 + k2 ) ⋅ n ) + i η ) 1 , (( k1 + k2 ) 2 − meta u0b (( k1 + k2 ) ⋅ nb ) + i η ) 1 , (( k1 + k2 ) 2 + ( k1 + k2 ) ⋅ ( 2 gkin meta u0b n − meta u0b nb ) − 2 gkin meta 2 u0b 2 + i η ) 1 }
FCLoopFindMomentumShifts: You can try to trade them for purely quadratic propagators using FCLoopReplaceQuadraticEikonalPropagators. \text{FCLoopFindMomentumShifts:
}\;\text{You can try to trade them for purely quadratic propagators
using FCLoopReplaceQuadraticEikonalPropagators.} FCLoopFindMomentumShifts: You can try to trade them for purely quadratic propagators using FCLoopReplaceQuadraticEikonalPropagators.
FCLoopFindMomentumShifts: Failed to derive the momentum shifts between topologies mytopo79 and mytopo67. This can be due to the presence of nonquadratic propagators or because shifts in external momenta are also necessary. \text{FCLoopFindMomentumShifts:
}\;\text{Failed to derive the momentum shifts between topologies
mytopo79 and mytopo67. This can be due to the presence of nonquadratic
propagators or because shifts in external momenta are also
necessary.} FCLoopFindMomentumShifts: Failed to derive the momentum shifts between topologies mytopo79 and mytopo67. This can be due to the presence of nonquadratic propagators or because shifts in external momenta are also necessary.
FCLoopFindTopologyMappings: Found 0 mapping relations \text{FCLoopFindTopologyMappings:
}\;\text{Found }0\text{ mapping relations } FCLoopFindTopologyMappings: Found 0 mapping relations
FCLoopFindTopologyMappings: Final number of independent topologies: 2 \text{FCLoopFindTopologyMappings:
}\;\text{Final number of independent topologies: }2 FCLoopFindTopologyMappings: Final number of independent topologies: 2
However, if we tell the code how some eikonal propagators can be
brought into a quadratic form, then an explicit mapping can be found
= FCLoopReplaceQuadraticEikonalPropagators[{ topoEik1, topoEik2}, -> { k1, k2}, -> { [ SPD[ k1 - k2]] -> SPD[ k1 - k2], [ SPD[ k1 + k2]] -> SPD[ k1 + k2]}, -> { SPD[ n ] -> 0 , SPD[ nb] -> 0 , SPD[ n , nb] -> 2 }] ;```mathematica eikMappings =
FCLoopFindTopologyMappings[toposNew];
```mathematica
FCLoopFindTopologyMappings: Found 1 mapping relations \text{FCLoopFindTopologyMappings:
}\;\text{Found }1\text{ mapping relations } FCLoopFindTopologyMappings: Found 1 mapping relations
FCLoopFindTopologyMappings: Final number of independent topologies: 1 \text{FCLoopFindTopologyMappings:
}\;\text{Final number of independent topologies: }1 FCLoopFindTopologyMappings: Final number of independent topologies: 1